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1.____________________________________________

Solar Energy SystemDesign

The largest solar electric generating plant in the world produces a maximum of354 megawatts (MW) of electricity and is located at Kramer Junction, California.This solar energy generating facility, shown below, produces electricity for theSouthern California Edison power grid supplying the greater Los Angeles area.The authors' goal is to provide the necessary information to design suchsystems.

The solar collectors concentrate sunlight to heat a heat transfer fluid to a hightemperature. The hot heat transfer fluid is then used to generate steam thatdrives the power conversion subsystem, producing electricity. Thermal energystorage provides heat for operation during periods without adequate sunshine.

Figure 1.1 One of nine solar electric energy generating systems at Kramer Junction, California,with a total output of 354 MWe. (photo courtesy Kramer Junction Operating Company)

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Another way to generate electricity from solar energy is to use photovoltaic cells;magic slivers of silicon that converts the solar energy falling on them directly intoelectricity. Large scale applications of photovoltaic for power generation, eitheron the rooftops of houses or in large fields connected to the utility grid arepromising as well to provide clean, safe and strategically sound alternatives to

current methods of electricity generation.

Figure 1.2 A 2-MW utility-scale photovoltaic power system co-located with a defunct nuclearpower plant near Sacramento, California.(photo courtesy of DOE/NREL, Warren Gretz)

The following chapters examine basic principles underlying the design andoperation of solar energy conversion systems such as shown in Figure 1.1 and1.2. This includes collection of solar energy, either by a thermal or photovoltaicprocess, and integration with energy storage and thermal-to-electric energyconversion to meet a predefined load. Study of the interaction of thesesubsystems yields the important guidelines for the design of optimal solar energy

systems. System design tools are provided to produce optimal sizing of bothcollector field and storage so that optimum system designs can be produced.

Since our emphasis is on the design of entire solar energy conversion systemsrather than design of its individual components, both thermal and photovoltaicsystems are included. This novel approach results from recognition of thecommonality of most system design considerations for both types of solar energysystems. We will not dwell on the intricacies of individual component design, but

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instead encourage the designer to take experimental (or predicted) componentinput/output information and incorporate this into an overall system design.

The system shown in Figure 1.1 employs parabolic trough line-focus collectors.We will cover this and other types of collectors for capturing the sun's energy

including flat plate, parabolic dish, central receiver and photovoltaic collectors.The purpose of a solar collector is to intercept and convert a reasonably largefraction of the available solar radiation. For solar thermal systems this energy isconverted into thermal energy at some desired temperature and then, maybe,into electricity.

For photovoltaic systems as shown in Figure 1.2, intercepted solar energy isconverted directly into low voltage direct current electricity. The engineeringtradeoff between cost and performance of the components necessary to performthese processes has led to a wide variety of collector and system designs.Reviews of solar collector designs representative of the different concepts that

have been built and tested are presented here.

The following sections serve as an overview of the solar energy system designprocess. They follow in a general manner, the flow of logic leading from the basicsolar resource to the definition of an operating solar energy conversion systemthat meets a specified demand for either thermal or electrical energy.

1.1 The Solar Energy Conversion System

There are many different types of solar energy systems that will convert the solarresource into a useful form of energy. A block diagram showing three of the most

basic system types is shown as Figure 1.3. In the first diagram , the solarresource is captured and converted into heat which is then supplied to a demandfor thermal energy (thermal load) such as house heating, hot water heating orheat for industrial processes. This type of system may or may not include thermalstorage, and usually include an auxiliary source of energy so that the demandmay be met during long periods with no sunshine.

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Figure 1.3 Diagram of a basic solar energy conversion systems. The AUX. box represents someauxiliary source of thermal or electrical energy.

If the demand (load) to be met is electricity (an electrical load) rather than heat,there are two common methods of converting solar energy into electricity. Onemethod is by collecting solar energy as heat and converting it into electricityusing a typical power plant or engine; the other method is by using photovoltaiccells to convert solar energy directly into electricity. Both methods are shown

schematically in Figure 1.3.

In general, if solar energy conversion systems are connected to a large electricaltransmission grid, no storage or auxiliary energy supply is needed. If the solarenergy conversion system is to be the only source of electricity, storage andauxiliary energy supply are usually both incorporated. If the thermal route ischosen, storage of heat rather than electricity may be used to extend theoperating time of the system. Auxiliary energy may either be supplied either asheat before the power conversion system, or as electricity after it. If thephotovoltaic route is chosen, extra electricity may be stored, usually in storagebatteries, thereby extending the operating time of the system. For auxiliary

power, an external electricity source is the only choice for photovoltaic systems.

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1.2 The Solar Resource

The basic resource for all solar energy systems is the sun. Knowledge of thequantity and quality of solar energy available at a specific location is of primeimportance for the design of any solar energy system. Although the solar

radiation (insolation) is relatively constant outside the earth's atmosphere, localclimate influences can cause wide variations in available insolation on the earthssurface from site to site. In addition, the relative motion of the sun with respect tothe earth will allow surfaces with different orientations to intercept differentamounts of solar energy.

Figure 1.4 shows regions of high insolation where solar energy conversionsystems will produce the maximum amount of energy from a specific collectorfield size. However, solar energy is available over the entire globe, and only thesize of the collector field needs to be increased to provide the same amount ofheat or electricity as in the shaded areas. It is the primary task of the solar

energy system designer to determine the amount, quality and timing of the solarenergy available at the site selected for installing a solar energy conversionsystem.

Figure 1.4 Areas of the world with high insolation.

Just outside the earth's atmosphere, the sun's energy is continuously available atthe rate of 1,367 Watts on every square meter facing the sun. Due to the earth's

rotation, asymmetric orbit about the sun, and the contents of its atmosphere, alarge fraction of this energy does not reach the ground. In Chapter 2, we discussthe effects of the atmospheric processes that modify the incoming solar energy,how it is measured, and techniques used by designers to predict the amount ofsolar energy available at a particular location, both instantaneously and over along term.

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As an example of the importance of the material discussed in Chapter 2, Figure1.5 shows the variation of insolation over a full, clear day in March at Daggett,California, a meteorological measurement site close to the Kramer Junction solarpower plant described previously. The outer curve, representing the greatest rateof incident energy, shows the energy coming directly from the sun (beam normal

insolation) and falling on a square meter of surface area which is pointed towardthe sun. The peak rate of incident solar energy occurs around 12:00 noon and is1,030 Watts per square meter. Over the full day, 10.6 kilowatt-hours of energyhas fallen on every square meter of surface area as represented by the areaunder this curve.

Figure 1.5 Insolation data from Daggett, California on a clear March day.

The middle curve represents the rate of solar energy falling on a horizontalsurface at the same location. For reasons to be discussed later this curveincludes both the energy coming directly from the sun's disc, and also thatscattered by the molecules and particles in the atmosphere (total horizontalinsolation). This scattered energy is shown as the bottom curve (diffuseinsolation). Over the entire day, 6.7 kilowatt-hours of solar energy fall on every

square meter of horizontal surface, of which 0.7 kilowatt-hours comes from alldirections other than directly from the sun.

Techniques for estimating the temporal solar resource at any site on the face ofthe earth are presented in Chapter 2. In addition, the development and use ofcomputerized meteorological data files is described. These data files based onlong-term actual observations, form the time-dependent database of the

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computerized performance computations contained within this book and, indeed,much of the solar literature.

An example of a complete set of beam normal insolation data for a given locationis shown in Figure 1.6. Here we see hourly insolation data, summarized over a

day, for each month of a year. With this type of data for a specific site, it ispossible to predict accurately the output of a solar energy conversion system,whether it is a low temperature thermal system, a high temperature thermalsystem or a photovoltaic system.

Figure 1.6 Time and date description of the global, horizontal insolation solar resource for CairoEgypt.

In addition to estimating the amount of energy coming from the sun, the solardesigner must also be able to predict the position of the sun. The sun's positionmust be known to predict the amount of energy falling on tilted surfaces, and todetermine the direction toward which a tracking mechanism must point acollector. Chapter 3 discusses the computation of the position of the sun withrespect to any given point on the face of the earth. Using only four parameters(latitude, longitude, date and local time), equations are derived to determine thelocation of the sun in the sky.

A characteristic fundamental to the capture of solar energy is that the amount ofenergy incident on a collector is reduced by a fraction equal to the cosine of theangle between the collector surface and the sun's rays. Knowing the position ofthe collector (or any other surface for that matter) and the position of the sunequations in Chapter 3 may be used to predict the fraction of incoming solarenergy that falls on the collector. These include situations where the collector isfixed or is tracked about a single axis, no matter what the orientation.

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1.3 Solar Collectors

The solar collector is the key element in a solar energy system. It is also thenovel technology area that requires new understandings in order to makecaptured solar energy a viable energy source for the future.

The function of a solar collector is simple; it intercepts incoming insolation andchanges it into a useable form of energy that can be applied to meet a specificdemand. In the following text, we will develop analytical understandings of flat-plate and concentrating collectors, as used to provide heat or electricity. Eachtype is introduced below.

Flat-plate thermal solar collectors are the most commonly used type of solarcollector. Their construction and operation are simple. A large plate of blackenedmaterial is oriented in such a manner that the solar energy that falls on the plateis absorbed and converted to thermal energy thereby heating the plate. Tubes or

ducting are provided to remove heat from the plate, transferring it to a liquid orgas, and carrying it away to the load. One (or more) transparent (glass or plastic)plates are often placed in front of the absorber plate to reduce heat loss to theatmosphere. Likewise, opaque insulation is placed around the backside of theabsorber plate for the same purpose. Operating temperatures up to 125oC aretypical.

Flat plate collectors have the advantage of absorbing not only the energy comingdirectly from the disc of the sun (beam normal insolation) but also the solarenergy that has been diffused into the sky and that is reflected from the ground.Flat plate thermal collectors are seldom tracked to follow the sun's daily path

across the sky, however their fixed mounting usually provides a tilt toward thesouth to minimize the angle between the sun's rays and the surface at noontime.Tilting flat-plate collectors toward the south provides a higher rate of energy atnoontime and more total energy over the entire day. Figure 1.7 shows aninstallation of flat-plate thermal collectors.

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Figure 1.7 Flat-plate thermal solar collectors for providing hot water.(photo courtesy of DOE/NREL,Warren Gretz)

Flat-plate photovoltaic collectors contain an array of individual photovoltaic cells,connected in a series/parallel circuit, and encapsulated within a sandwichstructure with the front surface being glass or plastic. Solar energy falls directlyupon the photovoltaic cell front surface and produces a small direct currentvoltage, providing electrical energy to a load. Unlike thermal collectors however,the backside of the panel is not insulated. Photovoltaic panels need to loose asmuch heat as possible to the atmosphere to optimize their performance.

Like flat-plate thermal collectors, flat-plate photovoltaic collectors (panels) absorbboth energy coming directly from the sun's disc, and diffuse and reflected energycoming from other directions. In general, flat-plate photovoltaic panels are

mounted in a fixed position and tilted toward the south to optimize noontime anddaily energy production. However, it is common to see flat-plate photovoltaicpanels mounted on mechanisms that track the sun about one tilted axis, therebyincreasing the daily output of the panels.

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Figure 1.8 Flat-plate photovoltaic collector applications.(photos courtesy of DOE/NREL, Warren Gretz)

When higher temperatures are required, concentrating solar collectors are used.Solar energy falling on a large reflective surface is reflected onto a smaller areabefore it is converted into heat. This is done so that the surface absorbing the

concentrated energy is smaller than the surface capturing the energy andtherefore can attain higher temperatures before heat loss due to radiation andconvection wastes the energy that has been collected. Most concentratingcollectors can only concentrate the parallel insolation coming directly from thesun's disk (beam normal insolation), and must follow (track) the sun's path acrossthe sky. Four types of solar concentrators are in common use; parabolic troughs(as used in the Kramer Junction, California solar energy electricity generatingplant shown in Figure 1.1), parabolic dishes, central receivers and Fresnellenses. Figure 1.9 shows these concepts schematically.

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Figure 1.9 Three commonly used reflecting schemes for concentrating solar energy to attainhigh temperatures.

A parabolic trough concentrates incoming solar radiation onto a line running the

length of the trough. A tube (receiver) carrying heat transfer fluid is placed alongthis line, absorbing concentrated solar radiation and heating the fluid inside. Thetrough must be tracked about one axis. Because the surface area of the receivertube is small compared to the trough capture area (aperture), temperatures up to400oC can be reached without major heat loss. Figure 1.10c shows one parabolictrough from the Kramer Junction, California field shown in Figure 1.1.

A parabolic dish concentrates the incoming solar radiation to a point. An insulatedcavity containing tubes or some other heat transfer device is placed at this pointabsorbing the concentrated radiation and transferring it to a gas. Parabolicdishes must be tracked about two axes. Figure 1.10b shows six 9kWe parabolic

dish concentrators with Stirling engines attached to the receiver at the focus.

A central receiver system consists of a large field of independently movable flatmirrors (heliostats) and a receiver located at the top of a tower. Each heliostatmoves about two axes, throughout the day, to keep the sun's image reflectedonto the receiver at the top of the tower. The receiver, typically a vertical bundleof tubes, is heated by the reflected insolation, thereby heating the heat transferfluid passing through the tubes. Figure 1.10a shows the 10 MWe Solar Onecentral receiver generating plant at Daggett, California with its adjoining steampower plant.

A Fresnel lens concentrator, such as shown in Figure 1.10d uses refraction ratherthan reflection to concentrate the solar energy incident on the lens surface to apoint. Usually molded out of inexpensive plastic, these lenses are used inphotovoltaic concentrators. Their use is not to increase the temperature, but toenable the use of smaller, higher efficiency photovoltaic cells. As with parabolicdishes, point-focus Fresnel lenses must track the sun about two axes.

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Figure 1.10a A central receiver system. (courtesy of Sandia National Laboratories, Albuquerque)

Figure 1.10b Two-axis tracking parabolic dish collectors. (courtesy of Schlaich, Bergermann und Partner)

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Figure 1.10c A single-axis tracking parabolic trough collector. (courtesy of Kramer Junction OperatingCompany)

Figure 1.10d A concentrating photovoltaic collector using Fresnel lenses. (courtesy of Amonix Corp.)

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1.4 Need for Storage

Like with any other power plant, solar power plant output must satisfy thedemands of the utility market. During peak demand periods, kilowatt-hour pricesare high and financial incentives are high for guaranteed supply. Solar plant input

is limited by diurnal, seasonal and weather-related insolation changes. In order tocope with these fluctuations, the solar plant input may be backed up by fossilfuels, or the solar changes may be mitigated by a buffering storage system. Thechoice depends on demands, system and site conditions, the relationshipbetween storage capacity and collector area is discussed in Chapter 10.

In thermal solar power plants, thermal storage and/or fossil backup act as:

an output management tool to prolong operation after sunset, to shift

energy sales from low revenue off-peak hours to high revenue peakdemand hours, and to contribute to guaranteed output

An internal plant buffer, smoothing out insolation charges for steadying

cycle operation, and for operational requirements such as blanketingsteam production, component pre-heating and freeze protection.

Photovoltaic plants in general need no internal buffer, and output managementcan be achieved with battery or other electrochemical storage, pumpedhydroelectric storage, or with diesel-generator backup. The implications ofbattery storage are discussed in Chapter 10.

Figure 1.11 Stored solar energy provides a firm capacity of 31MW until midnight at which timefossil fuel backup us used.

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1.5 Integration with Power Cycles

Because of their thermal nature, all the solar thermal technologies can behybridized, or operated with fossil fuel as well as solar energy. Hybridization hasthe potential to increase the value of concentrating solar thermal technology by

increasing its availability and dispatchability, decreasing its cost (by making moreeffective use of power generation equipment), and reducing technological risk byallowing conventional fuel use when needed.

Although an interconnected field of solar thermal collectors and thermal energystorage may be sufficient for providing high temperature heat directly to a thermaldemand, a power generation subsystem must be incorporated into the systemdesign if mechanical work or electrical power is to be an output from the system.Chapter 11 reviews the technology for power generation with particular emphasison power generation units suitable for interfacing with solar thermal energycollection subsystems. The inclusion of power generation in a solar thermal

energy design presents a challenge in selecting the appropriate designconditions. The efficiency of a power generation unit usually increases with theoperating temperature of the power generation cycle, whereas the efficiency ofsolar collectors decreases with temperature. A tradeoff must be performed todetermine the best system design point.

Figure 1.12 One of the steam cycle power cycles at the Kramer Junction solar energygenerating system.(photo courtesy of DOE/NREL, Warren Gretz)

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1.6 Site Qualification

Solar technologies using concentrating systems for electrical production requiresufficient beam normal radiation, which is the beam radiation which comes fromthe sun and passes through the planet's atmosphere without deviation and

refraction. Consequently, appropriate site locations are normally situated in aridto semi-arid regions. On a global scale, the solar resource in such regions is veryhigh. More exactly, acceptable production costs of solar electricity typically occurwhere radiation levels exceed about 1,700 kWh/m-yr, a radiation level found inmany areas as illustrated in Figure 1.4. Appropriate regions include thesouthwest United States, northern Mexico, the north African desert, the Arabianpeninsula, major portions of India, central and western Australia, the highplateaus of the Andean states, and northeastern Brazil. Promising site locationsin Europe are found in southern Spain and several Mediterranean islands.

Figure 1.13 A View of Kuraymat (Egypt), the envisaged site for a solar thermal power plant in the

Egyptian desert with cooling water from the Nile and connections to the national high voltagegrid.

Solar electricity generation costs and feasibility of the project highly depend onthe project site itself. A good site has to have a high annual beam insolation toobtain maximum solar electricity output. It must be reasonably flat toaccommodate the solar field without prohibitive expensive earth works. It must

also be close to the grid and a substation to avoid the need to build expensiveelectricity lines for evacuating the power. It needs sufficient water supply to coverthe demand for cooling water of its steam cycle. A backup fuel must be availablefor granting firm power during the times when no solar energy is available.Access roads must be suitable for transporting the heavy equipment like turbinegenerators to the site. Skilled personnel must be available to construct andoperate the plants. Chapter 13 reviews the criteria, methodology and examplesof site selection and qualification for solar plants.

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1.7 Economic and EnvironmentalConsiderations

The most important factor driving the solar energy system design process is

whether the energy it produces is economical. Although there are factors otherthan economics that enter into a decision of when to use solar energy; i.e. nopollution, no greenhouse gas generation, security of the energy resource etc.,design decisions are almost exclusively dominated by the levelized energy cost.This or some similar economic parameter, gives the expected cost of the energyproduced by the solar energy system, averaged over the lifetime of the system.In the following chapters, we will provide tools to aid in evaluating the factors thatgo into this calculation.

Commercial applications from a few kilowatts to hundreds of megawatts are nowfeasible, and plants totaling 354 MW have been in operation in California since

the 1980s. Plants can function in dispatchable, grid-connected markets or indistributed, stand-alone applications. They are suitable for fossil-hybrid operationor can include cost-effective storage to meet dispatchability requirements. Theycan operate worldwide in regions having high beam-normal insolation, includinglarge areas of the southwestern United States, and Central and South America,Africa, Australia, China, India, the Mediterranean region, and the Middle East, .Commercial solar plants have achieved levelized energy costs of about 12-15/kWh, and the potential for cost reduction are expected to ultimately lead tocosts as low as 5/kWh.

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Figure 1.14 Projections of levelized electricity cost predictions for large scale solar thermal powerplants. Current costs are shown in blue with a 1-2 cent/kWh addition for 'green' power shown ingreen.

1.8 Summary

The authors' overall objective is to illustrate the design of solar energy systems,both thermal and photovoltaic types. To do this, we examine the solar resourceand the ability of various types of solar collectors to capture it effectively. Designtools are developed which integrate performance of isolated solar collectors,along with energy storage, into a larger system that delivers either electrical orthermal energy to a demand. We show as many examples as possible, bothgraphic and photographic of these systems and their components.

It is our hope that once the simplicity of solar energy system design is

understood, engineers and manufacturers will provide new system designs thatwill expand the solar market worldwide and permit all to benefit from this clean,sustainable and distributed source of energy.

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3. The Suns PositionIn order to understand how to collect energy from the sun, one must first be able

to predict the location of the sun relative to the collection device. In this chapterwe develop the necessary equations by use of a unique vector approach. Thisapproach will be used in subsequent chapters to develop the equations for thesuns position relative to a fixed or tracking solar collector, (Chapter 4) and thespecial case of a sun-tracking mirror reflecting sunlight onto a fixed point(Chapter 10). Once developed, the sun position expressions of this chapter areused to demonstrate how to determine the location of shadows and the design ofsimple sundials. In outline form, our development looks like this:

o Earth-sun angles

o

Time

o Standard time zones

o Daylight savings time

o Sidereal time

o Hour angle

o Solar time

o Equation of timeo Time conversion

o Declination angle

o Latitude angle

o Observer-Sun Angles

o Solar altitude, zenith and azimuthangles

o Geometric view of suns path

o Daily and seasonal events

o Shadows and Sundials

Simple shadows Sundials

o Notes on Transformation of Vector Coordinates

o Summary

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Although many intermediate steps of derivation used to obtain the equationsdescribed in this chapter have been omitted, it is hoped that there are adequatecomments between steps to encourage the student to perform the derivation,thereby enhancing understanding of the materials presented. Brief notes on thetransformation of vector coordinates are included as Section 3.5 and a summary

of sign conventions for all of the angles used in this chapter is given in Table 3.3at the end of this chapter. Figures defining each angle and an equation tocalculate it are also included.

One objective in writing this chapter has been to present adequate analyticalexpressions so that the solar designer is able to develop simple computeralgorithms for predicting relative sun and collector positions for exact designconditions and locations. This will eliminate the need to depend on charts andtables and simplified equations.

3.1 Earth-Sun Angles

The earth revolves around the sun every 365.25 days in an elliptical orbit, with amean earth-sun distance of 1.496 x 1011 m (92.9 x 106 miles) defined as oneastronomical unit (1 AU). This plane of this orbit is called the ecliptic plane. Theearth's orbit reaches a maximum distance from the sun, or aphelion, of1.52 1011 m (94.4 106 miles) on about the third day of July. The minimum earth-sundistance, the perihelion, occurs on about January 2nd, when the earth is 1.47 1011 m (91.3 106 miles) from the sun. Figure 3.1 depicts these variations inrelation to the Northern Hemisphere seasons.

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Figure 3.1 The ecliptic plane showing variations in the earth-sun distance and the equinoxes andsolstices. The dates and day numbers shown are for 1981 and may vary by 1 or 2 days.

The earth rotates about its own polar axis, inclined to the ecliptic plane by 23.45degrees, in approximately 24-hour cycles. The direction in which the polar axispoints is fixed in space and is aligned with the North Star (Polaris) to within about45 minutes of arc (13 mrad). The earths rotation about its polar axis producesour days and nights; the tilt of this axis relative to the ecliptic plane produces our

seasons as the earth revolves about the sun.

3.1.1 Time

We measure the passage of time by measuring the rotation of the earth about itsaxis. The base for time (and longitude) measurement is the meridian that passesthrough Greenwich, England and both poles. It is known as the Prime Meridian.

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Today, the primary world time scale, Universal Time (previously called GreenwichMean Time), is still measured at the Prime Meridian. This is a 24-hour timesystem, based on mean time, according to which the length of a day is 24 hoursand midnight is 0 hours.

Mean timeis based on the length of an average day. A mean second is l/86,400of the average time between one complete transits of the sun, averaged over theentire year. In fact, the length of any one specific day, measured by the completetransit of the sun, can vary by up to 30 seconds during the year. The variable daylength is due to four factors listed in order of decreasing importance (Jespersonand Fitz-Randolph, 1977):

The earths orbit around the sun is not a perfect circle but elliptical, so the

earth travels faster when it is nearer the sun than when it is farther away.

The earths axis is tilted to the plane containing its orbit around the sun.

The earth spins at an irregular rate around its axis of rotation.

The earth `wobbles on its axis.

Standard Time Zones - Since it is conventional to have 12:00 noon beapproximately in the middle of the day regardless of the longitude, a system oftime zones has been developed. See (Blaise 2000) for an interesting story abouthow this unification developed. These are geographic regions, approximately 15degrees of longitude wide, centered about a meridian along which local standardtime equals mean solar time. Prior to about 1880, different cities (and even train

stations) had their own time standards, most based on the sun being due southat 12:00 noon.

Time is now generally measured about standard time zone meridians. Thesemeridians are located every 15 degrees from the Prime Meridian so that localtime changes in 1-hour increments from one standard time zone meridian to thenext. The standard time zone meridians east of Greenwich have times later thanGreenwich time, and the meridians to the west have earlier times.

Ideally, the meridians 7 degrees on either side of the standard time zonemeridian should define the time zone. However, boundaries separating time

zones are not meridians but politically determined borders following rivers,county, state or national boundaries, or just arbitrary paths. Countries such asSpain choose to be on `European Time (15o E) when their longitudes are wellwithin the adjacent Standard Time Meridian (0o). Figure 3.2 shows these timezone boundaries within the United States and gives the standard time zonemeridians (called longitudes of solar time).

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Figure 3.2 Time zone boundaries within the United States. From Jesperson and Fitz-Randolph1977.

Daylight Savings Time -To complicate matters further in trying to correlate clocktime with the movement of the sun, a concept known as daylight savings timewas initiated in the United States in the spring of 1918 to "save fuel and promoteother economies in a country at war" (Jesperson and Fitz-Randolph, 1977).According to this concept, the standard time is advanced by 1 hour, usually from2:00 AM on the first Sunday in April until 2:00 AM on the last Sunday in October.Although various attempts have been made to apply this concept uniformly withinthe country, it is suggested that the designer check locally to ascertain thecommitment to this concept at any specific solar site.

Sidereal Time - So far, and for the remainder of this text, all reference to time isto mean time, a time system based on the assumption that a day (86,400seconds) is the average interval between two successive times when a givenpoint on the earth faces the sun. In astronomy or orbital mechanics, however, theconcept ofsidereal time is often used. This time system is based on the siderealday, which is the length of time for the earth to make one complete rotation aboutits axis.

The mean day is about 4 minutes longer than the sidereal day because the earth,during the time it is making one revolution about its axis, has moved somedistance in its orbit around the sun. To be exact, the sidereal day contains 23hours, 56 minutes, and 4.09053 seconds of mean time. Since, by definition, thereare 86,400 sidereal seconds in a sidereal day, the sidereal second is slightlyshorter than the mean solar second is. To be specific: 1 mean second =1.002737909 sidereal seconds. A detailed discussion of this and other timedefinitions is contained in another work (Anonymous, 1981, Section B).

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3.1.2 The Hour Angle

To describe the earth's rotation about its polar axis, we use the concept of the

hour angle . As shown in Figure 3.3, the hour angle is the angular

distance between the meridian of the observer and the meridian whose planecontains the sun. The hour angle is zero at solar noon (when the sun reaches itshighest point in the sky). At this time the sun is said to be due south (or duenorth, in the Southern Hemisphere) since the meridian plane of the observercontains the sun. The hour angle increases by 15 degrees every hour.

Figure 3.3 The hour angle .This angle is defined as the angle between the meridianparallel to sun rays and the meridian containing the observer.

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Solar Time - Solar time is based on the 24-hour clock, with 12:00 as the timethat the sun is exactlydue south. The concept of solar time is used in predictingthe direction of sunrays relative to a point on the earth. Solar time is location(longitude) dependent and is generally different from local clock time, which isdefined by politically defined time zones and other approximations.

Solar time is used extensively in this text to define the rotation of the earthrelative to the sun. An expression to calculate the hour angle from solar time is

(3.1)

where ts is the solar time in hours.

EXAMPLE: When it is 3 hours after solar noon, solar time is 15:00 and the hour angle has avalue of 45 degrees. When it is 2 hours and 20 minutes before solar noon, solar time is 9:40 and

the hour angle is 325 degrees (or 35 degrees).

The difference between solar time and local clock time can approach 2 hoursat various locations and times in the United States, For most solar designpurposes, clock time is of little concern, and it is appropriate to present data interms of solar time. Some situations, however, such as energy demandcorrelations, system performance correlations, determination of true south, andtracking algorithms require an accurate knowledge of the difference betweensolar time and the local clock time.

Knowledge of solar time and Universal Time has traditionally been important toship navigators. They would set their chronometers to an accurately adjusted

tower clock visible as they left port. This was crucial for accurate navigation. Atsea a ship's latitude could be easily ascertained by determining the maximumaltitude angle of the sun or the altitude angle of Polaris at night. However,determining the ships longitude was more difficult and required that an accurateclock be carried onboard. If the correct time at Greenwich, England (or any otherknown location) was known, then the longitude of the ship could be found bymeasuring the solar time onboard the ship (through sun sightings) andsubtracting from it the time at Greenwich. Since the earth rotates through 360degrees of longitude every 24 hours, the ship then has traveled 1 degree oflongitude away from the Prime Meridian (which passes through Greenwich) forevery 4 minutes of time difference. An interesting story about developing

accurate longitude measurements may be found in Sobel, 1999.

Equation of Time - The difference between mean solar time and true solar timeon a given date is shown in Figure 3.4. This difference is called the equation oftime (EOT). Since solar time is based on the sun being due south at 12:00 noonon any specific day, the accumulated difference between mean solar time andtrue solar time can approach 17 minutes either ahead of or behind the mean,with an annual cycle.

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The level of accuracy required in determining the equation of time will depend onwhether the designer is doing system performance or developing trackingequations. An approximation for calculating the equation of time in minutes isgiven by Woolf (1968) and is accurate to within about 30 seconds during daylighthours.

(3.2)

where the anglex is defined as a function of the day numberN

(3.3)

with the day number, Nbeing the number of days since January 1. Table 3.1has been prepared as an aid in rapid determination of values ofNfrom calendar

dates.

Table 3.1 Date-to-Day Number Conversion

Month Day Number,N Notes

January d

February d + 31

March d + 59 Add 1 if leap year

April d + 90 Add 1 if leap year

May d + 120 Add 1 if leap year

June d + 151 Add 1 if leap year

July d + 181 Add 1 if leap year

August d + 212 Add 1 if leap year

September d + 243 Add 1 if leap year

October d + 273 Add 1 if leap year

November d + 304 Add 1 if leap year

December d + 334 Add 1 if leap year

Days of Special Solar Interest

Solar Event Date Day Number,NVernal equinox March 21 80

Summer solstice June 21 172

Autumnal equinox September 23 266

Winter solstice December 21 355NOTES:

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1. d is the day of the month2. Leap years are 2000, 2004, 2008 etc.

3. Solstice and equinox dates may vary by a day or two. Also, add 1 to thesolstice and equinox day number for leap years.

Figure 3.4 The equation of time (EOT). This is the difference between the local apparent solartime and the local mean solar time.

EXAMPLE: February 11 is the 42nd day of the year, therefore N = 42 and x is equal to 40.41

degrees, and the equation of time as calculated above is 14.35 minutes. This

compares with a very accurately calculated value of -14.29 minutes reported elsewhere(Anonymous, 198l). This means that on this date, there is a difference between the mean timeand the solar time of a little over 14 minutes or that the sun is "slow" relative to the clock by thatamount.

To satisfy the control needs of concentrating collectors, a more accuratedetermination of the hour angle is often needed. An approximation of theequation of time claimed to have an average error of 0.63 seconds and amaximum absolute error of 2.0 seconds is presented below as Equation (3.4)taken from Lamm (1981). The resulting value is in minutes and is positive whenthe apparent solar time is ahead of mean solar time and negative when the

apparent solar time is behind the mean solar time:

(3.4)

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Here n is the number of days into a leap year cycle with n = 1 being January 1of each leap year, and n =1461 corresponding to December 31 of the 4th year ofthe leap year cycle. The coefficients Ak and Bk are given in Table 3.2 below.Arguments for the cosine and sine functions are in degrees.

Table 3.2 Coefficients for Equation (3.4)

k Ak(hr) Bk(hr)

0 2.0870 10-4 0

1 9.2869 10-31.2229

10-1

25.2258

10-21.5698

10-1

31.3077

l0-35.1 602

10-3

42.1867

l0-32.9823

10-3

51.5100

10-42.3463

10-4

Time Conversion - The conversion between solar timeand clock time requires knowledge of the location, the day of the year, and thelocal standards to which local clocks are set. Conversion between solar time, tsand local clock time (LCT) (in 24-hour rather than AM/ PM format) takes the form

(3.5)

where EOT is the equation of time in minutes and LCis a longitude correctiondefined as follows:

(3.6)

and the parameterD in Equation (3.5) is equal to 1 (hour) if the location is in aregion where daylight savings time is currently in effect, or zero otherwise.

EXAMPLE: Let us find the clock time for solar noon at a location in Los Angeles, having alongitude of 118.3 degrees on February 11. Since Los Angeles is on Pacific Standard Time andnot on daylight savings time on this date, the local clock time will be:

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3.1.3 The Declination Angle

The plane that includes the earths equator is called the equatorial plane. If aline is drawn between the center of the earth and the sun, the angle between this

line and the earth's equatorial plane is called the declination angle ,asdepicted in Figure 3.5. At the time of year when the northern part of the earth'srotational axis is inclined toward the sun, the earths equatorial plane is inclined23.45 degrees to the earth-sun line. At this time (about June 21), we observe thatthe noontime sun is at its highest point in the sky and the declination angle

= +23.45 degrees. We call this condition the summer solstice, and it marks thebeginning of summer in the Northern Hemisphere.

As the earth continues its yearly orbit about the sun, a point is reachedabout 3 months later where a line from the earth to the sun lies on the equatorialplane. At this point an observer on the equator would observe that the sun wasdirectly overhead at noontime. This condition is called an equinox sinceanywhere on the earth, the time during which the sun is visible (daytime) isexactly 12 hours and the time when it is not visible (nighttime) is 12 hours. Thereare two such conditions during a year; the autumnal equinox on aboutSeptember 23, marking the start of the fall; and the vernal equinox on aboutMarch 22, marking the beginning of spring. At the equinoxes, the declination

angle iszero.

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Figure 3.5 The declination angle . The earth is shown in the summer solstice position

when = +23.45 degrees. Note the definition of the tropics as the intersection of the earth-sun line with the surface of the earth at the solstices and the definition of the Arctic and Antarcticcircles by extreme parallel sun rays.

The winter solstice occurs on about December 22 and marks the pointwhere the equatorial plane is tilted relative to the earth-sun line such that thenorthern hemisphere is tilted away from the sun. We say that the noontime sun isat its "lowest point" in the sky, meaning that the declination angle is at its most

negative value (i.e., = -23.45 degrees). By convention, winter declinationangles are negative.

Accurate knowledge of the declination angle is important in navigation andastronomy. Very accurate values are published annually in tabulated form in an

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ephemeris; an example being (Anonymous, 198l). For most solar designpurposes, however, an approximation accurate to within about 1 degree isadequate. One such approximation for the declination angle is

(3.7)

where the argument of the cosine here is in degrees andNis the day numberdefined for Equation (3.3) The annual variation of the declination angle is shownin Figure 3.5.

3.1.4 Latitude Angle

The latitude angle is the angle between a line drawn from a point onthe earths surface to the center of the earth, and the earths equatorial plane.

The intersection of the equatorial plane with the surface of the earth forms theequator and is designated as 0 degrees latitude. The earths axis of rotationintersects the earths surface at 90 degrees latitude (North Pole) and -90 degreeslatitude (South Pole). Any location on the surface of the earth then can bedefined by the intersection of a longitude angle and a latitude angle.

Other latitude angles of interest are the Tropic of Cancer (+23.45 degreeslatitude) and the Tropic of Capricorn (- 23.45 degrees latitude). These representthe maximum tilts of the north and south poles toward the sun. The other twolatitudes of interest are the Arctic circle (66.55 degrees latitude) and Antarcticcircle (-66.5 degrees latitude) representing the intersection of a perpendicular to

the earth-sun line when the south and north poles are at their maximum tiltstoward the sun. As will be seen below, the tropics represent the highest latitudeswhere the sun is directly overhead at solar noon, and the Arctic and Antarcticcircles, the lowest latitudes where there are 24 hours of daylight or darkness. Allof these events occur at either the summer or winter solstices.

3.2 Observer-Sun Angles

When we observe the sun from an arbitrary position on the earth, we areinterested in defining the sun position relative to a coordinate system based atthe point of observation, not at the center of the earth. The conventional earth-

surface based coordinates are a vertical line (straight up) and a horizontal planecontaining a north-south line and an east-west line. The position of the sunrelative to these coordinates can be described by two angles; the solar altitudeangle and the solar zenith angle defined below. Since the sun appears not as apoint in the sky, but as a disc of finite size, all angles discussed in the followingsections are measured to the center of that disc, that is, relative to the "centralray" from the sun.

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3.2.1 Solar Altitude, Zenith, and Azimuth Angles

The solar altitude angle is defined as the angle between the central rayfrom the sun, and a horizontal plane containing the observer, as shown in Figure3.6. As an alternative, the suns altitude may be described in terms of the solar

zenith angle which is simply the complement of the solar altitude angleor

(3.8)

The other angle defining the position of the sun is the solar azimuth angle (A).It is the angle, measured clockwise on the horizontal plane, from the north-pointing coordinate axis to the projection of the suns central ray.

Figure 3.6 Earth surface coordinate system for observer at Q showing the solar azimuth angle,

the solar altitude angle and the solar zenith angle for a central sun rayalong direction vectorS. Also shown are unit vectors i,j, kalong their respective axes.

The reader should be warned that there are other conventions for the solarazimuth angle in use in the solar literature. One of the more common

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conventions is to measure the azimuth angle from the south-pointing coordinaterather than from the north-pointing coordinate. Another is to consider thecounterclockwise direction positive rather than clockwise. The information inTable 3.3 at the end of this chapter will be an aid in recognizing these differenceswhen necessary.

It is of the greatest importance in solar energy systems design, to be able tocalculate the solar altitude and azimuth angles at any time for any location on theearth. This can be done using the three angles defined in Section 3.1 above;

latitude , hour angle , and declination . If the reader isnot interested in the details of this derivation, they are invited to skip directly tothe results; Equations (3.17), (3.18) and (3.19).

For this derivation, we will define a sun-pointing vector at the surface ofthe earth and then mathematically translate it to the center of the earth with adifferent coordinate system. Using Figure 3.6 as a guide, define a unit direction

vectorS pointing toward the sun from the observer location Q:

(3.9)

where i, j, and kare unit vectors along thez, e, and n axes respectively. Thedirection cosines ofSrelative to the z, e, andn axes are Sz, SeandSn, respectively.These may be written in terms of solar altitude and azimuth as

(3.10)

Similarly, a direction vector pointing to the sun can be described at the centerof the earthas shown in Figure 3.7. If the origin of a new set of coordinates isdefined at the earths center, the m axis can be a line from the origin intersectingthe equator at the point where the meridian of the observer at Qcrosses. Theeaxis is perpendicular to the m axis and is also in the equatorial plane. The thirdorthogonal axis p may then be aligned with the earths axis of rotation. A newdirection vectorSpointing to the sun may be described in terms of its directioncosines Sm , S'eand S'prelative to the m, e, and p axes, respectively. Writing these

in terms of the declination and hour angles, we have

(3.11)

Note that Seis negative in the quadrant shown in Figure 3.7.

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Figure 3.7 Earth center coordinate system for the sun ray direction vector S defined in terms of

hour angle and the declination angle .

These two sets of coordinates are interrelated by a rotation about the eaxis

through the latitude angle and translation along the earth radius QC. Wewill neglect the translation along the earths radius since this is about 1 / 23,525of the distance from the earth to the sun, and thus the difference between thedirection vectors SandS is negligible. The rotation from the m, e, pcoordinatesto the z, e, n coordinates, about the eaxis is depicted in Figure 3.8. Both sets ofcoordinates are summarized in Figure 3.9. Note that this rotation about the eaxisis in the negative sense based on the right-hand rule. In matrix notation, thistakes the form

(3.12)

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Figure 3.8 Earth surface coordinates after translation from the observer at Q to the earth centerC.

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Figure 3.9 Composite view of Figures 3.6, 3.7 and 3.8 showing parallel sun ray vectorsSandSrelative to the earth surface and the earth center coordinates.

Solving, we have

(3.13)

Substituting Equations (3.10) and(3.1)(3.11) for the direction cosine gives usour three important results

(3.14)

(3.15)

(3.16)

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Equation (3.14) is an expression for the solar altitude angle in terms of theobservers latitude (location), the hour angle (time), and the suns declination

(date). Solving for the solar altitude angle , we have

(3.17)

Two equivalent expressions result for the solar azimuth angle (A) from eitherEquation (3.15) or(3.16). To reduce the number of variables, we could substituteEquation (3.14) into either Equation (3.15) or (3.16); however, this substitutionresults in additional terms and is often omitted to enhance computational speedin computer codes.

The solar azimuth angle can be in any of the four trigonometric quadrantsdepending on location, time of day, and the season. Since the arc sine and arccosine functions have two possible quadrants for their result, both Equations

(3.15) and (3.16) require a test to ascertain the proper quadrant. No such test isrequired for the solar altitude angle, since this angle exists in only one quadrant.

The appropriate procedure for solving Equation (3.15) is to test the result todetermine whether the time is before or after solar noon. For Equation (3.15), atest must be made to determine whether the solar azimuth is north or south ofthe east-west line.

Two methods for calculating the solar azimuth (A), including the appropriatetests, are given by the following equations. Again, these are written for theazimuth angle sign convention used in this text, that is, that the solar azimuth

angle is measured from due north in a clockwise direction, as with compassdirections. Solving Equation (3.15) the untested result,A then becomes

(3.18)

A graphical description of this test will follow in the next section.

Solving Equation (3.16), the untested result,A"becomes

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(3.19)

In summary, we now have equations for both the suns altitude angle andazimuth angle written in terms of the latitude, declination and hour angles. Thisnow permits us to calculate the suns position in the sky, as a function of date,

time and location (N, , ).

EXAMPLE: For a site in Miami, Florida (25 degrees, 48 minutes north latitude/ 80degrees, 16 minutes west longitude) at 10:00 AM solar time on August 1 (not a leap year), find

the sun's altitude, zenith and azimuth angles.....For these conditions, the declination angle iscalculated to be 17.90 degrees, the hour angle -30 degrees and the sun's altitude angle is then61.13 degrees, the zenith angle 28.87 degrees and the azimuth angle 99.77 degrees.

3.2.2 A Geometric View of Suns Path

The path of the sun across the sky can be viewed as being on a disc displacedfrom the observer. This "geometric" view of the sun's path can be helpful invisualizing sun movements and in deriving expressions for testing the sun anglesas needed for Equation (3.18) to ascertain whether the sun is in the northern sky.

The sun may be viewed as traveling about a disc having a radiusR at a constantrate of 15 degrees per hour. As shown in Figure 3.10, the center of this discappears at different seasonal locations along the polar axis, which passes

through the observer at Q and is inclined to the horizon by the latitude angle

pointing toward the North Star (Polaris).

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Figure 3.10 A geometric view of the suns path as seen by an observer at Q. Each disc has

radiusR.

The center of the disc is coincident with the observerQ at the equinoxes and is

displaced from the observer by a distance of at other times of theyear. The extremes of this travel are at the solstices when the disc is displacedby 0.424 R along the polar axis. It can be seen that in the winter, much of thedisc is "submerged" below the horizon, giving rise to fewer hours of daylight andlow sun elevations as viewed from Q.

At the equinoxes, the sun rises exactly due east at exactly 6:00 AM (solartime) and appears to the observer to travel at a constant rate across the skyalong a path inclined from the vertical by the local latitude angle. Exactly one-half

of the disc is above the horizon, giving the day length as 12 hours. At noon, theobserver notes that the solar zenith angle is the same as the local latitude. Thesun sets at exactly 6:00 PM, at a solar azimuth angle of exactly 270 degrees ordue west.

In the summer, the center of the disc is above the observer, giving rise to morehours of daylight and higher solar altitude angles, with the sun appearing in thenorthern part of the sky in the mornings and afternoons.

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Since the inclination of the polar axis varies with latitude it can be visualized thatthere are some latitudes where the summer solstice disc is completely above thehorizon surface. It can be shown that this occurs for latitudes greater than 66.55degrees, that is, above the Arctic Circle. At the equator, the polar axis ishorizontal and exactly half of any disc appears above the horizon surface, which

means that the length of day and night is 12 hours throughout the year.

A test to determine whether the sun is in the northern part of the sky may bedeveloped by use of this geometry. Figure 3.11 is a side view of the suns disclooking from the east.

Figure 3.11 Side view of sun path disc during the summer when the disc centerYis above theobserver at Q.

In the summer the sun path disc of radiusR has its centerYdisplaced above theobserverQ. PointX is defined by a perpendicular from Q. In the n- z plane, the

projection of the position Sonto the line containingXand Ywill be

where is the hour angle. The appropriate test for the sun being in thenorthern sky is then

(3.20)

The distance XY can be found by geometry arguments. Substituting intoEquation (3.20), we have

(3.21)

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This is the test applied to Equation (3.18) to ensure that computed solarazimuth angles are in the proper quadrant.

3.2.3 Daily and Seasonal Events

Often the solar designer will want to predict the time and location of sunrise andsunset, the length of day, and the maximum solar altitude. Expressions for theseare easily obtained by substitutions into expressions developed in Section 3.2. l.

The hour angle for sunset (and sunrise) may be obtained from Equation (3.17) bysubstituting the condition that the solar altitude at sunset equals the angle to thehorizon. If the local horizon is flat, the solar altitude is zero at sunset and the hour

angle at sunset becomes

(3.22)

The above two tests relate only to latitudes beyond 66.55 degrees, i.e.

above the Arctic Circle or below the Antarctic Circle.

If the hour angle at sunset is known, this may be substituted into Equation(3.18) or(3.19) to ascertain the solar azimuth at sunrise or sunset.

The hours of daylight, sometimes of interest to the solar designer, may becalculated as

(3.23)

where s is in degrees. It is of interest to note here that although the hours of

daylight vary from month to month except at the equator (where ),there are always 4,380 hours of daylight in a year (non leap year) at any locationon the earth.

Another limit that may be obtained is the maximum and minimum noontime

solar altitude angle. Substitution of a value of into Equation (3.17)gives

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(3.24)

where denotes the absolute value of this difference. Aninterpretation of Equation (3.24) shows that at solar noon, the solar zenith angle(the complement of the solar altitude angle) is equal to the latitude angle at theequinoxes and varies by 23.45 degrees from summer solstice to winter solstice.

EXAMPLE: At a latitude of 35 degrees on the summer solstice (June 21), find the solar timeof sunrise and sunset, the hours of daylight, the maximum solar altitude and the compassdirection of the sun at sunrise and sunset.....On the summer solstice, the declination angle is+23.45 degrees, giving an hour angle at sunset of 107.68 degrees. Therefore, the time ofsunrise is 4:49:17 and of sunset, 19:10:43. There are 14.36 hours of daylight that day, and thesun reaches a maximum altitude angle of 78.45 degrees. The sun rises at an azimuth angle of60.94 degrees (north of east) and sets at an azimuth angle of 299.06 degrees (north of west).

Other seasonal maxima and minima may be determined by substituting theextremes of the declination angle, that is, 23.45 degrees into the previouslydeveloped expressions, to obtain their values at the solstices. Since the solardesigner is often interested in the limits of the suns movements, this tool can bevery useful in developing estimates of system performance.

3.3 Shadows and Sundials

Now that we have developed the appropriate equations to define the direction ofthe sun on any day, any time and any location, lets look at two applications,which are interesting, and may be of value to the solar designer.

3.3.1 Simple Shadows

An important use of your understanding of the suns position is in predicting thelocation of a shadow. Since sunlight travels in straight lines, the projection of anobscuring point onto the ground (or any other surface) can be described in termsof simple geometry.

Figure 3.12 shows a vertical pole on a horizontal surface. The problem here is todefine the length and direction of the shadow cast by the pole. This will first bedone in terms of radial coordinates and then in Cartesian coordinates.

In polar coordinates, for a pole height, OP and the shadow azimuth, s defined inthe same manner as the previous azimuth angles i.e. relative to true north withclockwise being positive, we have for the shadow length:

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(3.25)

and for the shadow azimuth:

(3.26)

In terms of Cartesian coordinates as shown on Figure 3.12, with the base ofthe pole as the origin, north as the positive y-direction and east the positive x-direction, the equations for the coordinates of the tip of the shadow from thevertical pole OP are:

(3.27)

(3.28)

Figure 3.12 Shadow cast by pole OP showing x and y coordinates of shadow tip, and shadowazimuth (As).

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Shadow coordinates are important for performing solar studies, especially relatedto the siting of solar energy systems. All that is needed is the latitude, hour angleand declination (i.e. the date) along with the location of the point causing theshadow. With a little bit of imagination, these equations can be used to predictshadowing of nearby terrain by trees or structures. All that is needed is to define

a set of points on an object, and predict the shadow cast by that point

Example: At a latitude of 35.7 degrees north on August 24 (day number 236) at 14:00, the sunsaltitude is 53.1 degrees and its azimuth 234.9 degrees. Then the shadow cast by a 10 cm highvertical pole is 7.51 cm long, and has a shadow azimuth (from north) of 54.9 degrees. The x-and y-coordinates of the tip of the shadow are: x = 6.14 cm and y = 4.32 cm.

3.3.1 Sundials

Throughout the history of mankind, the suns shadow has been used to tell hetime of day. Ingenious devices casting shadows on surfaces of all shapes andorientations have been devised. However, all of these can be understood with asimple manipulation of Equations (3.25) through (3.28) developed above.

The most common sundial consists of a flat, horizontal cardwith lines markedfor each hour of daylight. These lines spread out from a point with the noon linepointing toward the North Pole. A gnomonor shadow-casting device,is mountedperpendicular to the card and along the noon line. The gnomon is tapered at the

local latitude angle, with its height increasing toward the north. Figure3.13 below shows a simple horizontal sundial with the important angles defined.

Figure 3.13 The simple horizontal sundial

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In order to derive an equation for the directions of the time lines on the card, it

must be assumed that the declination angle is zero (i.e. at theequinoxes). Using Equations (3.27) and (3.28) to define the coordinates of theshadow cast by the tip of the gnomon, and Equations (3.14), (3.15) and (3.16) to

describe these in terms of the latitude angle , and the hour angle

with the declination angle being set to zero. An expression for the anglebetween the gnomon shadow and its base may be derived. Through simple

geometric manipulations, the expression for this angle simplifies to

(3.29)

In making a horizontal sundial, it is traditional to incorporate a wise sayingabout time or the sun, on the sundial card. An extensive range of sources ofinformation about sundials, their history, design and construction is available onthe internet.

3.4 Notes on the Transformation of Vector Coordinates

A procedure used often in defining the angular relationship between a surfaceand the sun rays is the transformation of coordinates. Defining the vectorV, weobtain

(3.30)

where i, j, and k are unit vectors collinear with the x, y, and z-axes,respectively, and the scalar coefficients (Vx, Vy, Vz)are defined along these sameaxes. To describe V in terms of a new set of coordinate axes x', y ', and z'withtheir respective unit vectors i', j' and k'in the form

(3.31)

we must define new scalar coefficients . Using matrixalgebra will do this. A column matrix made of the original scalar coefficientsrepresenting the vector in the original coordinate system is multiplied by a matrixwhich defines the angle of rotation.

The result is a new column matrix representing the vector in the newcoordinate system. In matrix form this operation is represented by

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(3.32)

whereAijis defined in Figure 3.14 for rotations about thex, y,orzaxis.

Figure 3.14Axis rotation matrices AI j for rotation about the three principal coordinate axes by the

angle .

It should be noted here that if the magnitude of V is unity, the scalarcoefficients in Equations (3.30) and (3.31)are direction cosines. Note also that

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the matrices given in Figure 3-22 are valid only for right-handed, orthogonal

coordinate systems, and when positive values of the angle of rotation, are determined by the right-hand rule.

Summary

Table 3.3 summarizes the angles described in this chapter along with their zerovalue orientation, range and sign convention.

Table 3.3 Sign Convention for Important Angles

Title Symbol ZeroPositiveDirection

RangeEquationNo.

Figure No.

Earth-Sun Angles

Latitude equator northernhemisphere

+/- 90o --- 3.3

Declination equinox summer +/- 23.45o (3.7) 3.5

Hour Angle noon afternoon +/- 180o (3.1) 3.3

Observer-Sun Angles

Sun Altitude horizontal upward 0 to 90o (3.17) 3.6

Sun Zenith vertical toward horizon 0 to 90o (3.8) 3.6

Sun Azimuth A due north clockwise* 0 to 360o(3.18) or(3.19)

3.6

Shadow and Sundial Angles

ShadowAzimuth

As due north clockwise 0 to 360o (3.26) 3.12

GnomonShadow

noon afternoon +/- 180o (3.29) 3.13

* looking down + axis toward origin

References and Bibliography

Anonymous (1981), "The Astronomical Almanac for the Year 1981," issued by theNautical Almanac Office of the United States Naval Observatory.

Blaise, C. (2000), Time Lord - Sir Sandford Fleming and the Creation of StandardTime, Pantheon Books, New York.

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Jesperson, J., and J. Fitz-Randolph (1977), "From Sundials to Atomic Clocks,"National Bureau of Standards Monograph 155, December.

Lamm, L. 0. (1981), "A New Analytic Expression for the Equation of Time," SolarEnergy26 (5), 465.

Sobel, D. (1995), Longitude - The True Story of a Lone Genius Who Solved theGreatest Scientific Problem of His Time ,Fourth Estate Ltd., London.

Woolf, H. M. (1968), "On the Computation of Solar Evaluation Angles and theDetermination of Sunrise and Sunset Times, "National Aeronautics and SpaceAdministration Report NASA TM-X -164, September.

Zimmerman, J. C. (1981), "Sun Pointing Programs and Their Accuracy," SandiaNational Laboratories Report SAND81-0761, September.

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4.______________________________Collecting Solar Energy

In Chapter 2, we developed an understanding of how to determine the rate andamount of energy coming from the sun. We introduced in that section theconcept of the cosine effectorcosine loss representing the difference betweenthe amount of energy falling on a surface pointing at the sun, and a surfaceparallel to the surface of the earth.

In order to collect solar energy here on the earth, it is important to know the anglebetween the suns rays and a collector surface (aperture). When a collector is notpointing (or more exactly, when the collector aperture normal is not pointing)directly at the sun, some of the energy that could be collected is being lost.

In this chapter, we develop the equations to calculate the angle between acollector aperture normal and a central ray from the sun. This development isdone first for fixed and then for tracking collectors. These equations are thenused to provide insight into collector tracking and orientation design by predictingthe integrated solar radiation energy that is incident on the collector aperture andcould be collected. These concepts will be developed with the following outline:

o Aperture-sun angles: the angle of

incidence Fixed (non-tracking) apertures

Single-axis tracking apertures Horizontal tracking axis Tilted tracking axis Offset aperture Vertical tracking axis Vertical tracking axis with

offset aperture Tracking axis tilted at

latitude angle Two-axis tracking apertures

Azimuth / elevation

tracking Polar (equatorial) tracking

o Collector aperture irradiance and solar

radiation energy Irradiance on a collector aperture

Direct (beam) apertureirradiance

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Global (total) apertureirradiance

Solar radiation energy on acollector aperture

Examples for clear days

Two-axis tracking apertures Single-axis tracking

apertures Fixed apertures

Examples using TMY database

o Algorithm for aperture irradiance

o SHADOW - A simple model for collector

field shadowing

Parabolic trough and flat-platecollector fields Parabolic dish fields

o Summary

4.1 Aperture-Sun Angles; The Angle of Incidence

In Chapter 3, we defined the sun's position angles relative to earth-center

coordinates and then to coordinates at an arbitrary location onthe earths surface ( ) and (A) and a functional relationship between theseangles, i.e., Equations (3.13), (3.14) and (3.15). In the design of solar energysystems, it is most important to be able to predict the angle between the sunsrays and a vector normal (perpendicular) to the aperture or surface of the

collector. This angle is called the angle of incidence . Knowing thisangle is of critical importance to the solar designer, since the maximum amountof solar radiation energy that could reach a collector is reduced by the cosine ofthis angle.

The other angle of importance, discussed in this section is the tracking

angle, . Most types of mid- and high-temperature collectors require atracking drive system to align at least one axis and often two axes of a collectoraperture normal to the suns central ray. The tracking angle is the amount ofrotation required to do this.

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